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Parikh vectors
2EBI_1 7AOL_1 4RCW_1 Letter Amino acid
4 24 7 T Threonine
3 17 8 A Alanine
8 11 9 R Arginine
6 9 10 E Glutamic acid
3 7 5 H Histidine
13 11 13 K Lycine
2 14 7 Q Glutamine
8 16 18 S Serine
1 27 13 V Valine
4 20 26 N Asparagine
1 12 8 C Cysteine
5 30 39 L Leucine
1 13 11 P Proline
3 3 3 W Tryptophan
2 11 7 Y Tyrosine
6 17 14 D Aspartic acid
5 26 11 G Glycine
2 11 14 I Isoleucine
5 10 6 M Methionine
4 17 11 F Phenylalanine 

2EBI_1|Chain A|DNA binding protein GT-1|Arabidopsis thaliana (3702)
>7AOL_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4RCW_1|Chains A, B|SLIT and NTRK-like protein 1|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
2EBI , Knot 49 86 0.84 40 75 84 
KKRAETWVQDETRSLIMFRRGMDGLFNTSKSNKHLWEQISSKMREKGFDRSPDMCTDKWRNLLKEFKKAKHHDRGNGSAKMSYYKE
7AOL , Knot 135 306 0.84 40 196 289 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGLVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
4RCW , Knot 104 240 0.79 40 160 223 
CPGGCSCDHIPGSGLKMNCNNRNVSSLADLKPKLSNVQELFLRDNKIHSIRKSHFVDYKNLILLDLGNNNIATVENNTFKNLLDLRWLYMDSNYLDTLSREKFAGLQNLEYLNVEYNAIQLILPGTFNAMPKLRILILNNNLLRSLPVDVFAGVSLSKLSLHNNYFMYLPVAGVLDQLTSIIQIDLHGNPWECSCTIVPFKQWAERLGSEVLMSDLKCETPVNFFRKDFMLLSNDEICPQ

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(2EBI_1)}(2) \setminus P_{f(7AOL_1)}(2)|=40\), \(|P_{f(7AOL_1)}(2) \setminus P_{f(2EBI_1)}(2)|=161\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:00010011000000111100110111000000001100100010001100010100001001100100100000101010100000
Pair \(Z_2\) Length of longest common subsequence
2EBI_1,7AOL_1 201 3
2EBI_1,4RCW_1 167 3
7AOL_1,4RCW_1 176 4

Newick tree

 
[
	7AOL_1:97.83,
	[
		2EBI_1:83.5,4RCW_1:83.5
	]:14.33
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{392 }{\log_{20} 392}-\frac{86}{\log_{20}86})=94.4\)
Status Protein1 Protein2 d d1/2
Query variables 2EBI_1 7AOL_1 120 77
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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